The product of the first seven positive integers is equal to \[7 \times 6\times 5\times 4\times 3\times 2\times 1= 5040\] Mathematicians will write this product as \(7!\). This is read as “\(7\) factorial”. So, \(7! = 7 \times 6\times 5\times 4\times 3\times 2\times 1= 5040\).
This factorial notation can be used with any positive integer. For example, \(11! = 11 \times 10 \times 9 \times \cdots \times 3 \times 2 \times 1 = 39\,916\,800\). The three dots “\(\cdots\)” represent the product of the integers between \(9\) and \(3\).
Suppose \(N = 1000!\). That is, \[N = 1000! = 1000 \times 999 \times 998\times 997 \times \cdots \times 3\times 2\times 1\]
Note that \(N\) is divisible by \(5\), \(25\), \(125\), and \(625\). Each of these factors is a power of \(5\). That is, \(5=5^1\), \(25 = 5^2\), \(125= 5^3\), and \(625 = 5^4\).
Determine the largest power of \(5\) that divides \(N\).